We call a minimum cost restricted time combinatorial optimization (MCRT) problem any problem that has a finite set P , finite family S of subsets of P , nonnegative threshold h, and two non-negative real-valued functions y : P!R+ (say, cost) and x : P!R+ (say, time). One seeks a solution F 2 S with y(F ) = minfy(F ) : F 2 S; x(F ) hg, where z(G) = P g2G z(g) for every z 2 fx; yg and G 2 S. We also assume that for the corresponding minimum cost problem there is an efficient exact or approximation algorithm. We describe an approximation algorithm for any MCRT problem. Though our algorithm is not polynomial in general, we provide some theoretical and practical evidence that the algorithm may be fairly fast in many cases.

We give a generalization of binomial queues involving an arbitrary sequence (mk )k=0;1;2;::: of integers greater than one. Different sequences lead to different worst case bounds for the priority queue operations, allowing the user to adapt the data structure to the needs of a specific application. Examples include the first priority queue to combine a sub-logarithmic worst case bound for Meld with a sub-linear worst case bound for Delete min. Keywords: Data structures; Meldable priority queues. 1 Introduction The binomial queue, introduced in 1978 by Vuillemin [14], is a data structure for meldable priority queues. In meldable priority queues, the basic operations are insertion of a new item into a queue, deletion of the item having minimum key in a queue, and melding of two queues into a single queue. The binomial queue is one of many data structures which support these operations at a worst case cost of O(logn) for a queue of n items. Theoretical [2] and empirical [9] evidence i...

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